Tessellations

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Tessellations
Lindsay Stone Fara Mandelbaum Sara
Fazio Cori McGrail Laura Welch
Jason Miller
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Definition
Tessellation a careful juxtaposition of
elements into a coherent pattern
sometimes called a mosaic or tiling
Example
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Tessellations are different from patterns
because patterns usually do not have distinct
closed shapes A closed shape is a shape that has
a definite interior and a definite exterior
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History

• Mathematics
• Johannes Kepler 1619
• Russian crystallographer E.S. Fedorov 1891

Science -X-ray Crystallography
This picture is a transformation of eight points
in an array to make a very small crystal
lattice which tessellates
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Science Continued.
This image suggests the relationship between
tessellations,symmetry, and X-ray crystallography
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M.C. Escher (1898 1972)
-Created over 100 tessellated patterns
-Work involves topology, optical
illusions, hyperbolic
tessellations, and other
advanced mathematical topics
-Eschers tilings were designed to resemble
recognizable objects
-Eschers work with tilings of the plane
embodies many ideas that scientists and
mathematicians discovered only after Escher did
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Sun and Moon

• Uses birds to transform day into night
• In this image the white birds bring forth the sun
  while the dark birds carry the moon and the
  stars. Day and night fight each other for
  attention but fit seemlessly together.

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Symmetry Drawing No. 71

• Symmetry Drawing No. 71 is one of his most
  complex with 12 different birds forming a
  rectangle in this image

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Regular Tessellations
-A regular polygon tessellation is constructed
from regular polygons
-Regular polygons have equal sides and equal
angles
-The regular polygons must fill the plane at each
vertex, with repeating patterns and no
overlapping pieces
Note This pentagon does not fit the
vertextherefore it is not a regular tessellation
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There are only 3 regular tessellations
One of triangles
One of squares
One of hexagons
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This is NOT a regular polygon tessellation
because..
The plane is not filled at the vertex because
there is a space left over
vertex
space
A regular polygon tessellation, can be changed by
using alterations to the sides of the
polygon. These alterations are called
transformations
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Three Common Transformations

• 1. Translation which is a slide of one
• side of the polygon, move
• 2. Reflection flip or mirror image of
• one side of the polygon
• 3. Rotation turn of a side around one
• vertex of a polygon

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Translation slide
this side
moves here
the alteration
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Reflections flip
the alteration
flips here
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Rotation turn
the alteration
here
rotates around this vertex
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Steps to name an arrangement of regular
polygons around a vertex

• To name an arrangement of regular polygons around
  a vertex,
• first find the regular polygon with the least
  number of sides.
• 2. Then find the longest consecutive run of this
  polygon, that is, two or
• more repetitions of this polygon around the
  vertex.
• 3. Next, indicate the number of sides of this
  regular polygon. For
• example, to name a triangle with 3 sides, we name
  it 3 and follow it
• with a period (.). If you find more than one
  consecutive "run" of this
• polygon, then name it twice, i.e., 3.3.
• 4. Proceeding in a clockwise or counterclockwise
  order, indicate the
• number of sides of each polygon as you see them
  in the arrangement.
• 5. Do remember to start with the longest
  consecutive run of the
• regular polygon with the shortest number of
  sides.

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Semi-regular Tessellations
Definition are tessellations of more than one
type of regular polygons such that the polygon
arrangement at each vertex is the same






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In order for the semi-regular tessellation to
work, the interior angle sum must be equal to 360
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Semi-Regular Tessellations
3.12.12
4.6.12
4.8.8
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Semi-Regular Tessellations
3.4.6.4
3.6.3.6
3.3.3.3.6
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Semi-Regular Tessellations
3.3.3.4.4
3.3.4.3.4
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Demi-regular Tessellations
Definition tessellations of regular polygons in
which there are two or three different polygon
arrangements
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Duals and Vertex Configurations
Duals - connect the centers of the regular
polygons around a vertex creating a new
shape Vertex Configurations connect the
midpoints of the sides of the regular polygons
around a vertex creating a new shape
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In life tessellations appear all around us.
Mud Flats
Honeycombs
Hydrogen Peroxide
Checkers
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Gallery

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References

• Totally Tessellated - ThinkQuest winner - great
  site, instruction, information.
• http//library.advanced.org/16661/

• Tessellations Tutorials - Math Forum site
• http//forum.swarthmore.edu/sum95/suzanne/tess.int
  ro.html - site for construction of
  tessellations. http//forum.swarthmore.edu/sum95/s
  uzanne/links.html - great list of tessellation
  links

• Math. Com - List of good tessellation links
• http//test.math.com/students/wonders/tessellation
  s.html

• World of Escher site - commerical site with
  gallery of Dutch artist,Escher who was famous for
  his tessellation art.
• http//WorldOfEscher.com/gallery/

• Science Universitys Tilings Around Us Site.
  http//www.ScienceU.com/geometry/articles/tiling/t
  ilings.html

• Other links from Forum.
• http//forum.swarthmore.edu/library/topics/transfo
  rm_g/